Universal gravitation 1-determine height of satellite

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SUMMARY

The discussion revolves around calculating the height of a second satellite orbiting Earth using Kepler's Laws. The first satellite has a period of 1.5 hours and orbits 250 km above Earth's surface. The second satellite has a period of 7.5 hours. By applying Kepler's Third Law, the correct calculation for the height of the second satellite above Earth's surface is determined to be approximately 2,618.29 km. The initial miscalculation of 80,970,595 km was corrected after realizing the need to take the cube root in the calculations.

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Homework Statement



Two satellites are orbiting around the Earth. One satellite has a period of 1.5 h and is 250 km above the Earth's surface. The second satellite has a period of 7.5 h. Using Kepler's Laws and the fact that the Earth's radius is 6.38x106 m determine the height of the second satellite above the Earth's surface.

Homework Equations



Kepler's 3rd law: (Ta/Tb)2=(Ra/Rb)3

motion of planets must conform to circular motion equation: Fc=4∏2mR/T2

From Kepler's 3rd law: R3/T2=K or T2=R3/K

Gravitational force of attraction between the sun and its orbiting planets: F=(4∏2Ks)*m/R2=Gmsm/R2

Gravitational force of attraction between the Earth and its orbiting satelittes: F=(4∏2Ke)m/R2=Gmem/R2

Newton's Universal Law of Gravitation: F=Gm1m2/d2

value of universal gravitation constant is: G=6.67x10-11N*m2/kg2

weight of object on or near Earth: weight=Fg=mog, where g=9.8 N/kg
Fg=Gmome/Re2

g=Gme/(Re)2

determine the mass of the Earth: me=g(Re)2/G

speed of satellite as it orbits the Earth: v=√GMe/R, where R=Re+h

period of the Earth-orbiting satellite: T=2∏√R3/GMe

Field strength in units N/kg: g=F/m

Determine mass of planet when given orbital period and mean orbital radius: Mp=4∏2Rp3/GTp2


The Attempt at a Solution



So for satellite #1 we have,
T1=1.5 hr
h1=250 km above Earth's surface

For second satellite we have,
T2=7.5hr
h2=?

We know that the Earth's radius is RE=6.38x106 m

so R1=250000m+ (6.38x106m)= 6630000m

R2=RE+h2

I used (Ta/Tb)2=(Ra/Rb)3
to solve for R2=8.098x1010m

Then I did, R2-RE=h2
h2=80 970 595 km

Could someone please just verify that what I did here is correct? and if I made any mistakes if someone could please point them out to me that would be greatly appreciated! Thanks so much in advance !
 
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dani123 said:

Homework Statement



Two satellites are orbiting around the Earth. One satellite has a period of 1.5 h and is 250 km above the Earth's surface. The second satellite has a period of 7.5 h. Using Kepler's Laws and the fact that the Earth's radius is 6.38x106 m determine the height of the second satellite above the Earth's surface.

Homework Equations



Kepler's 3rd law: (Ta/Tb)2=(Ra/Rb)3

[STRIKE]motion of planets must conform to circular motion equation: Fc=4∏2mR/T2

From Kepler's 3rd law: R3/T2=K or T2=R3/K

Gravitational force of attraction between the sun and its orbiting planets: F=(4∏2Ks)*m/R2=Gmsm/R2

Gravitational force of attraction between the Earth and its orbiting satelittes: F=(4∏2Ke)m/R2=Gmem/R2

Newton's Universal Law of Gravitation: F=Gm1m2/d2

value of universal gravitation constant is: G=6.67x10-11N*m2/kg2

weight of object on or near Earth: weight=Fg=mog, where g=9.8 N/kg
Fg=Gmome/Re2

g=Gme/(Re)2

determine the mass of the Earth: me=g(Re)2/G

speed of satellite as it orbits the Earth: v=√GMe/R, where R=Re+h

period of the Earth-orbiting satellite: T=2∏√R3/GMe

Field strength in units N/kg: g=F/m

Determine mass of planet when given orbital period and mean orbital radius: Mp=4∏2Rp3/GTp2

[/STRIKE]

The Attempt at a Solution



So for satellite #1 we have,
T1=1.5 hr
h1=250 km above Earth's surface

For second satellite we have,
T2=7.5hr
h2=?

We know that the Earth's radius is RE=6.38x106 m

so R1=250000m+ (6.38x106m)= 6630000m

R2=RE+h2

I used (Ta/Tb)2=(Ra/Rb)3
to solve for R2=8.098x1010m

Then I did, R2-RE=h2
h2=80 970 595 km

Could someone please just verify that what I did here is correct? and if I made any mistakes if someone could please point them out to me that would be greatly appreciated! Thanks so much in advance !
I crossed out a bunch of stuff you don't need for this problem.

You have definitely done something wrong! The distance from Sun to Earth is only about 1.5×108km .

The ratio of T2/T1 = 5.

The ratio of R2/R1 should be less than 5.

Did you forget to take the cube root in getting your answer?
 
You are absolutely right, i didn't put it in my calculator correctly... so this time i got R2=8998.29 km or 8998294.4m

but from here, how do i go about getting the value for h2 which is ultimately what the question is asking for?
 
I proceeded to do the exact thing as I did in my attempt above and got h2=2618.29 km

Does this seem reasonable?

Thanks again so much for your help!
 
dani123 said:
You are absolutely right, i didn't put it in my calculator correctly... so this time i got R2=8998.29 km or 8998294.4m

but from here, how do i go about getting the value for h2 which is ultimately what the question is asking for?
I don't think that's right.

Show your steps in detail.
 
I did:

R1=(250000m)+(6.38x106m)= 6 630 000m

Then I went on to find R2=Re+h

So with the equation (T1/T2)2=(R1/R2)3

(R2)3= (R1)3/(T1/T2)2

R2=(6 630 000m)3/(1.5/7.5)2= 8998294.399 m

And then I manipulated the R=Re+h equation and solved for h=2618294.399 m
 
Last edited:
The problem asks for the height of the satellite above the Earth's surface. Aren't you calculating the distance from the Earth's center?
 
sorry, yes your right, I just fixed it... does that seem right?
 
does h= 2618294.399m seem like the correct answer?
 

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